WASHINGTON — Hours after their seven-year pledge to dismantle the Affordable Care Act hurtled off the rails in the Senate, Republican lawmakers pointed fingers at their own on Friday for letting their voters down. Democrats exulted in blocking the repeal effort, at least for now.

For House Republicans — who in May passed a bill, at no small political peril to several vulnerable members, to upend President Barack Obama’s signature domestic achievement — the message was clear: We did our job. And the Senate must not give up.

“They’ve got to get back at it,” said Representative Tom MacArthur, Republican of New Jersey.

“They’ve got to keep voting,” said Representative Thomas Massie, another Kentucky Republican, going a step further to suggest that Senate Republicans who stood in the way of the repeal should be removed at the ballot box.

“I think the voters need to do a little more sorting of the people who are serving up here,” he said.

The Democratic leaders in both chambers, Senator Chuck Schumer of New York and Representative Nancy Pelosi of California, urged Republicans to abandon the push for a repeal once and for all and join them in patching up problems in the existing health law.

“Right now we go forward recognizing the value of the Affordable Care Act,” Ms. Pelosi told reporters on Friday. “But we also know that there are updates and improvements we need to make.”

President Trump has suggested little appetite for this approach.

Early Friday morning, just after the vote, he seemed to reiterate a preference for allowing the health law to falter under the expectation that Democrats’ leverage will be weakened down the line, causing them to embrace right-leaning remedies.

“As I said from the beginning, let Obamacare implode, then deal,” Mr. Trump tweeted shortly after the vote. “Watch!”

Several Republicans have raised concerns with this strategy. “I don’t want to cause hardship for a lot of people,” said Representative Charlie Dent of Pennsylvania, who voted against the House bill in May. “Bipartisan plans are out there.”

Hours after his initial message, Mr. Trump returned to another idea he has floated: doing away with the legislative filibuster in the Senate, which requires 60 votes on most legislation. He tweeted this strategy even though the health law was considered under circumstances that required a simple majority.

“If Republicans are going to pass great future legislation in the Senate, they must immediately go to a 51 vote majority, not senseless 60,” he wrote on Twitter. “Even though parts of health care could pass at 51, some really good things need 60. So many great future bills & budgets need 60 votes.”

Some Republicans, left to grapple with the fallout of the repeal failure, opted instead for collective self-flagellation.

Representative Brian Mast, Republican of Florida, was asked if lawmakers might face consequences in next year’s midterm elections if they could not keep their promise to undo the law.

“Everybody should,” he said. “If we don’t get the No. 1 job that we said that we would do done, yeah, people should be held accountable.”

Then there was the scene at a meeting of Republican House members on Friday morning.

According to lawmakers, the gathering included a recitation of lyrics from “The Wreck of the Edmund Fitzgerald,” a song about a sinking ship, which was likened to the Senate’s stumble.